By Dan Laksov

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**Example text**

In other words, we have that ψ∗ is a functor from presheaves, respectively sheaves, on X to presheaves, respectively sheaves, on Y . For all x in X we have a canonical map of stalks:!! 1) that takes the class of the pair (V, t), where V is an open neighbourhood of ψ(x) and t ∈ ψ∗ (F )(V ) = F (ψ −1 (V )), to the class in Fx of the pair (ψ −1 (V ), t). It is clear that the map (ψF )x is independent of the choice of the representative (V, t) of the class tψ(x) . Let V be an open subset of Y and U an open subset of ψ −1 (V ).

Tn ] are prime ideals, and (t1 , t2 , . . , tn ) is maximal. 13) Proposition. Let a be an ideal in A. (1) The ideal a is prime if and only if the ring A/a is an integral domain. (2) The ideal a is maximal if and only if the ring A/a is a field. → n Proof. (1) Let ϕA/a : A → A/a be the canonical homomorphism. Then f is not in a if and only if ϕA/a (f ) is not zero in A/a. Hence assertion (1) follows from the equality ϕA/a (f )ϕA/a (g) = ϕA/a (f g) valid for all pairs of elements f, g of A. (2) We have that a is maximal if and only if Af + a = A for all elements f ∈ / a.

P in the ring A is a prime ideal if it is different from A, and if f, g are elements in A such that if f g ∈ p, then f ∈ p or g ∈ p. Equivalently we have that an ideal p is prime if for each pair of elements f, g in A that are not in p we have that f g is not in p. m of A is maximal if it is different from A and it is not contained in any ideal in A different from A and m. A ring with only one maximal ideal is called a local ring. We shall denote by m = mA the maximal ideal in a local ring A. A homomorphism ϕ : A → B of local rings is called local if it maps the maximal ideal in A to the maximal ideal in B, or equivalently if we have ϕ−1 (mB ) = mA .

### Algebra by Dan Laksov

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